# Paper «On partial CAP-subgroups of finite groups» published in J. Algebra

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Adolfo Ballester-Bolinches, Luis M. Ezquerro, Yangming Li, and Ning Su

On partial CAP-subgroups of finite groups

J. Algebra, 431 (2015), 196–208

http://dx.doi.org/10.1016/j.jalgebra.2015.01.035

Abstract

Given a chief factor H/K of a finite group G, we say that a subgroup A of G avoids H/K if H∩A=K∩A; if HA=KA, then we say that A covers H/K. If A either covers or avoids the chief factors of some given chief series of G, we say that A is a partial CAP-subgroup of G. Assume that G has a Sylow p-subgroup of order exceeding pk. If every subgroup of order pk, where k≥1, and every subgroup of order 4 (when pk=2 and the Sylow 2-subgroups are non-abelian) are partial CAP-subgroups of G, then G is p-soluble of p-length at most 1.

2010 Mathematics subject classification: 20D10; 20D20

Keywords: Finite group; Partial CAP-subgroup; p-soluble group; p-length

# Paper «On formations of finite groups with the generalised Wielandt property for residuals» published in J. Algebra

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A. Ballester-Bolinches, S. F. Kamornikov, and V. Pérez-Calabuig

On formations of finite groups with the generalised Wielandt property for residuals

J. Algebra., 412 (2014), 173–178

http://dx.doi.org/10.1007/s11856-013-0030-y

Abstract

A formation F of finite groups has the generalised Wielandt property for residuals, or F is a GWP-formation, if the F-residual of a group generated by two F-subnormal subgroups is the subgroup generated by their F-residuals. We prove that every GWP-formation is saturated. This is one of the crucial steps in the hunt for a solution of the classification problem.

2010 Mathematics subject classification: 20D10; 20D20

Keywords: finite group; formation; residual; subnormality

# Paper «On a problem of L.A. Shemetkov on superradical formations of finite groups» published in J. Algebra

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A. Ballester-Bolinches, S. F. Kamornikov, V. N. Tyutyanov

On a problem of L.A. Shemetkov on superradical formations of finite groups

J. Algebra, 403, 69-76 (2014)

http://dx.doi.org/10.1016/j.jalgebra.2014.01.004

Abstract: Subgroup-closed saturated formations F which are closed under taking products of F-subnormal F-subgroups are studied in the paper. Our results can be regarded as further developments in the hunt for a solution of a problem proposed by L.A. Shemetkov in 1999 in the Kourovka Notebook.

MSC: 20D10, 20D20
Keywords: Finite group, saturated formation, critical group

# Paper «Generalised norms in finite soluble groups» published in J. Algebra

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A. Ballester-Bolinches, John Cossey, Liangcai Zhang

Generalised norms in finite soluble groups

J. Algebra, 402, 392-405 (2014)

http://dx.doi.org/10.1016/j.jalgebra.2013.12.012

Abstract: We give a framework for a number of generalisations of Baer’s norm that have appeared recently. For a class C of finite nilpotent groups we define the C-norm κC(G) of a finite group G to be the intersection of the normalisers of the subgroups of G that are not in C. We show that those groups for which the C-norm is not hypercentral have a very restricted structure. The non-nilpotent groups G for which G = κC (G) have been classified for some classes. We give a classification for nilpotent classes closed under subgroups, quotients and direct products of groups of coprime order and show the known classifications can be deduced from our classification.

MSC: 20D10, 20D20, 20D25
Keywords: Finite group, normaliser, norm

# Paper «A note on Sylow permutable subgroups of infinite groups» published in J. Algebra

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A. Ballester-Bolinches, S. Camp-Mora, L. A. Kurdachenko

A note on Sylow permutable subgroups of infinite groups

J. Algebra, 398, 156-161 (2014)

http://dx.doi.org/10.1016/j.jalgebra.2013.08.042

Abstract: A subgroup A of a periodic group G is said to be Sylow permutable,
or S-permutable, subgroup of G if A P = P A for all Sylow subgroups
P of G. The aim of this paper is to establish the local nilpotency
of the section A^G /Core_G( A) for an S-permutable subgroup A of a
locally finite group G.
MSC: 20E15, 20F19, 20F22
Keywords: Locally finite group, Hyperfinite group, Sylow permutability, Ascendant subgroup